Chapter 10 Engineering Economics
10.1. Introduction
- Engineering problems usually can be solved in many different ways and choosing a
particular solution is dependent on various factors, among which money is the most
important.
- As an example, suppose you are an engineer setting up a new toy production plant in
HK. Then, there are various problems that are related to the money. For instance, you
can lease or buy the land, and you can make or purchase machines. These monetary
problems constitute the engineering economics
- Although engineering economics is not new, it is introduced to engineering students
(except industrial engineering students) only recently. And you are among the first
batch.
- The purpose of engineering economics is to study whether any project or investment
is financially desirable.
- Engineering economics problems are usually complicated, not because it is difficult
to calculate but because of the following reasons:
- The information / data is inaccurate / incomplete (e.g., the interest rate varies)
- The evaluation criteria are biased (e.g., what is your opinion of good computer)
- In this chapter, will study the basics of engineering economics including:
- The value of time
- The rate of return
- The payback period
- The utility theory
10.2. Compound Interest Formula
- Before we started out discussion, let us introduce several important concepts:
- The time value of money. It is clearly that one dollar ten years ago is not the same
as one dollar today. Hence, when involving money, time has a value and it is
referred to as the time value of money.
- The interest rate. When you borrow money from a bank (e.g., your credit card),
you pay interest. When you save your money in a bank, the bank pay you interest.
Interest is everywhere.
- The discount rate. The discount rate represents real change in value to a person or
group. It is different to the interest rate and is always greater than the interest rate.
- The minimum attractive rate of return. This is another interpretation of interest
rate.
- In order to study the interest, let us first introduce the following definition.
i = effective interest rate per unit time, which also represents the minimum attractive
rate of return (MARR)
n = number of compounding periods
P = present sum of money
F = future sum of money
A = end-of-period cash flow
G = uniform increase or decrease in end-of-period case flows or amounts.
- Based on the above notations, we can then draw the cash flow diagram:
P
F
A
1
A
2
A
3
G
A
n-1
2G
A
n
(n-2)G
nG
From the diagram, it is seen with an initial amount of F and paying amount of A per
time period in n periods, we have the following:
at the first period (n = 1), the money value is: P(1 + i)
at the second period (n = 2), the money value is: [P(1 + i)](1 + i) = P(1 + i)2
…
at the last period (n = n), the money value is: F = P(1 + i)n
Therefore,
F
 (1  i ) 2
P
this is called the single payment compound amount formula and is denoted as:
(F/P, i, n) = (1 + i)n
-
which represents the value of money in the future. Its inverse is the single payment
present worth formula:
1
( P / F , i, n) 
(1  i ) 2
Now let us suppose that amount of A is paid periodically, then the balance is:
at the first period: P(1 + i) – A
at the second period: [P(1 + i) – A](1 + i) – A = P(1 + i)2 – A(1 + i) – A
at the third period: P(1 + i)3 – A(1 + i)2 – A(1 + i) – A
…
at the last period: P(1 + i)n – A(1 + i)n-1 – A(1 + i)n-2 … A(1 + i) – A = 0
On the last equation, subtracting the same equation multiple by (1 + i), it can be
shown that:
(1  i) n  1
( P / A, i, n) 
i(1  i) n
-
-
-
this is called the uniform series present worth formula. Its inverse is the uniform
series capital recovery formula:
i (1  i) n
( A / P, i , n ) 
(1  i) n  1
Since F = P(1 + i)n, we can find the relationship between F and A, called the uniform
series compound amount formula:
(1  i) n  1
( F / A, i, n) 
i
and its inverse is the uniform series sinking fund formula:
i
( A / F , i, n) 
(1  i ) n  1
Example 1: suppose that a $20,000 piece of equipment is expected to last 5 years and
then result in a $4000 salvage value. If the interest rate of return is 15%, what are (a)
the annual equivalent cost and (b) the present equivalent cost?
(a) A = -$20,000(A/P,15,5) + $4,000(A/F,15,5)
= -$20,000(0.2983) + $4,000(0.1483)
= -$5,373
(b) P = -$20,000 + $4,000(P/F,15,5)
= -$20,000 + $4,000(0.4972)
= -$18,011
This example is the same as you buy a car.
Note:
- When you use the compound interest formulas, you can use the chain rule. For
example: P = F(P/F, i, n) = F(P/A, i, n) (A/F, i, n).
- You can account the effect of inflation onto the interest rate: suppose the inflation
rate is f, then the equivalent interest rate: ie = i + f + if.
- You can add a few point on the interest rate as the risk premium.
- The rate of return is the same as the interest rate: given P, F, n, the calculated i is
the rate of return.
- The payback period is the same as the total period: given P, F, i, the calculated n
is the payback period.
2.3. Comparison of Alternatives and the Equivalent Worth Method
- The other important issues in engineering economics is to choose among alternatives.
- Following is the procedure in comparing alternatives
Step 1: define the alternatives
Step 2: determine the study period
Step 3: provide estimates of the cash flows for each alternatives
Step 4: specify the interest rate
Step 5: select the measures of effectiveness
Step 6: compare the alternatives
Step 7: perform sensitivity analyses
Step 8: select preferred alternative
-
-
From the economic point of view, in order to compare the alternatives we need to
convert different items to a same base, for example the present value. This is the basic
idea of equivalent worth methods.
The following example shows the present worth method. Consider the following two
mutually exclusive alternatives and recommend which one should be implemented.
Assuming that the interest rate is 15% and the years of study is 10.
Alternatives
A
Initial cost
Life
Salvage value
Annual receipts
Annual disbursements
$20,000
5 years
$4,000
$10,000
$4,400
B
$30,000
10 years
0
$14,000
$8,600
Now we convert everything to the present value:
Alternatives
Initial cost
Salvage value at the end of year 10
(P/F, 15, 10) = 0.2475
Annual receipts
(P/A, 15,10) = 5.0188
Annual disbursements
(P/A, 15, 10) = 5.0188
Replacement
($20,000 - $4,000)(P/F,15,5)
Total
-
A
- $20,000
B
- $30,000
$989
0
$50,188
$70.263
- $22,083
- $43,162
- $7,955
$1,139
- $2,889
Therefore, Alternative A should be chosen.
Similar methods include the annual worth method, future worth method and the rate
of return method.
You may want to analyze your own financial situation using these methods.
2.4 Sensitivity Analysis
- Sensitivity analysis is used to study the relative magnitude of a particular variation in
one or several elements of a problem that is sufficient to alter a particular decision.
- Similarly, break-even analysis is used to determine the conditions where two
alternatives are equivalent.
- Following is an example: A project may be constructed to full capacity now or may
be undertaken in two stages. The construction costs are as follows:
- Two stages construction:
- Construct first stage: $100,000
- Construct second state n years from now $120,000
- Full capacity construction: $140,000
Other factors include:
- all facilities will last until 40 years from now regardless of when they are
installed; at that time, they will have zero salvage value
- the annual cost of operation and maintenance is the same for both alternatives
- the interest rate is 8%
Now let us use the present worth to study the two alternatives and determine when
they will break even. First, for the full capacity construction:
PW(full capacity) = $140,000
Second, the cost of the two stage construction alternative depends on when the second
state is constructed, (i.e., the value of n). Following is a list of values against different
n.
n=3
n = 10
n = 20
n = 30
PW(two stage) = 100,000 + 120,000(P/F, 8, 5) = $181,700
PW(two stage) = 100,000 + 120,000(P/F, 8, 10) = $155,600
PW(two stage) = 100,000 + 120,000(P/F, 8, 20) = $125,700
PW(two stage) = 100,000 + 120,000(P/F, 8, 30) = $111,900
It is seen that the break-even time is somewhere between 10 to 20 years, (in fact, it is
about 15 years).
Based on this calculation, if it is predicted that, owing to the demand, full capacity
will be needed within 10 years, then, we should chose to construct full capacity.
2.5 Utility theory
- It is not unusual that the investments are facing uncertain. In fact, there is always an
uncertainty (e.g., the stock market in Hong Kong today). To deal with the uncertainty,
the utility theory is developed.
- Suppose A is an action (or alternative), xj, j = 1, 2, …, n are and possible outcomes
and pj are the probabilities, then the expected monetary value of A is defined by the
utility function:
n
EMV ( A)   p j x j
j 1